I had the same problem last year and i queued up at Hitherfield Children's Centre. It takes ages as each birth registration took 20-30 minutes by the time they filled out all the forms (and they started late) so between turning up early to be sure of being seen and waiting for my slot (i was 6th in the queue) i was there for over 3 hours.
They had a milkspot at the same time so i was able to join that group while i waited so it worked out ok.
I just checked their website and it now says they take bookings so it might be worth calling them to see if you can get an appointment
https://fluencycontent-schoolwebsite.ne ... ations.pdf
Getting Irish passport afterwards was easy and they turned it around in a few days.
But FYI - you don't need passport for baby to travel with Aer Lingus between UK and Ireland (other airlines policy is less clear). It is clearly stated on their website
http://www.aerlingus.com/travelinformat ... passports/
Citizens of Ireland and the UK under the age of 16 do not need a photo ID if travelling with their parent/guardian but we recommend that they carry some form of identification (e.g. birth certificate, bus pass or student card) in case this is requested by the immigration authorities at entry.
Apparently nobody actually checks and you can use your hospital discharge papers if they do ask and you haven't got birth cert yet. So that's another potential solution for you.